On the Use of Flexible, Rectilinear Blocks
to Obtain Minimum-Area Floorplans in
Mixed Block and Cell Designs
DINESH P. MEHTA
University of Tennessee Space Institute
and
NAVEED SHERWANI
Intel Corporation
This paper presents three minimum-area floorplanning algorithms that use flexible arbitrary
rectilinear shapes for the standard cell regions in MBC designs. The first algorithm (pure
HCST) introduces a grid traversal technique which guarantees a minimum-area floorplan.
The second algorithm (Hybrid-BF) uses a combination of HCST and Breadth First (BF)
traversals to give a practical solution that approximately places flexible blocks at specified
locations called seeds. The third algorithm (Hybrid-MBF) improves on the shapes of the
flexible blocks generated by Hybrid-BF by using a combination of HCST and a Modified
Breadth First (MBF) traversal. All three algorithms are polynomial in the number of grid
squares. Optimized implementations of Hybrid-BF and Hybrid-MBF required less than two
seconds on a SUN SPARCstation10.
Categories and Subject Descriptors: B.7.2 [Integrated Circuits]: Design Aids—Layout; F.2.2
[Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—Routing and layout; G.2.2 [Discrete Mathematics]: Graph Theory—Graph algorithms
General Terms: Algorithms
Additional Key Words and Phrases: Floorplanning, mixed block and cell designs, rectilinear
polygons
1. INTRODUCTION
Traditional floorplanning algorithms have used rectangles [Kozminski and
Kinnen 1988; Otten 1982; Suthantavibul et al. 1991; Vijayan and Tsay
1991; Chong and Sahni 1993; Pan and Liu 1995; Dai et al. 1989; Pan et al.
Authors’ addresses: D. P. Mehta, Computer Science Department, University of Tennessee
Space Institute, Tullahoma, TN 37388-9700; N. Sherwani, Design Technologies, Intel Corporation, Hillsboro, OR 97124.
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ACM Transactions on Design Automation of Electronic Systems, Vol. 5, No. 1, January 2000, Pages 82–97.
Flexible, Rectilinear Blocks in Mixed Block and Cell Designs
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1994] and simple rectilinear shapes [Wang and Wong 1992; Dasgupta et al.
1995; Wong and Liu 1989; Yeap and Sarrafzadeh 1993; Nakatake et al.
1996] to realize placeable modules. The reason for this limitation was to
“facilitate automation” [Sarrafzadeh and Wong 1996]; i.e., to simplify the
algorithms themselves and to simplify subsequent place-and-route within
the modules. Rectangular floorplans were further limited to sliceable
floorplans to simplify the algorithms (e.g., the floorplan area optimization
problem is NP-complete for general floorplans, but solvable in polynomial
time for sliceable floorplans [Otten 1983; Stockemeyer 1983]). Sliceable
floorplans also simplify the channel ordering problem. However, these
limitations to rectangles and further, to sliceable floorplans led to restricted floorplan solutions. A recognition of this led to algorithms that also
use simple rectilinear shapes (e.g., L-shapes) to represent modules. These
approaches improve floorplan quality, but make the algorithms more
complex because of the additional cases that have to be considered.
In this paper we will demonstrate that the removal of these limitations
by using arbitrary rectilinear shapes results in practical algorithms which
produce zero-wasted-area floorplans. Moreover, current layout tools can
work on a rectilinear boundary by modeling it as a bounding rectangle with
keepout regions. Further, since 5– 6 routing layers are available in current
designs, routing is primarily accomplished in over-the-cell areas rather
than in channels. Thus, the floorplanner does not need to be concerned with
the channel ordering problem. Although our algorithms are general, they
are especially applicable to mixed block and cell (MBC) designs [Sherwani
1995]. In an MBC design, regular structures such as RAM and datapath
blocks are usually implemented as full-custom macro cells (called blocks or
fixed blocks). The remaining logic is implemented using a “sea of standard
cells.” Blocks are said to be “glued” together by these cells. A common
approach to laying out MBC designs consists of forming groups of standard
cells called flexible blocks or juice blocks. Our floorplanner assumes that a
seed location is specified for each flexible block. This specifies an approximate location for that block. In MBC designs, an expert (either a program
or a knowledgeable user) can exploit knowledge of the functional relationship between a flexible block and related fixed blocks to specify a good
location. Most of the existing layout algorithms for MBC designs require
flexible blocks to be rectangles [Apte and Kedem 1990; Upton et al. 1990;
Putatunda et al. 1988; Lee 1993]. All of these, except Lee [1993], restrict
the fixed block shapes to rectangles. In Lee [1993], the fixed block shapes
may be rectilinear, but flexible blocks are still rectangular.
We present three floorplanning algorithms, all of which utilize a grid
data representation of the floorplan area. The first algorithm (pure HCST)
employs a novel grid traversal technique, which guarantees a minimumarea floorplan. However, this algorithm places flexible blocks without
regard to their seed locations. Another flaw with pure HCST is that it
results in floorplans where flexible blocks have long snakelike shapes. Such
shapes are undesirable because they make it difficult for the subsequent
place-and-route step to converge. For example, routability would be a
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problem if a large number of nets must be routed through a long, narrow
region. Also, such shapes permit two cells in a flexible block to be very far
away from each other. Any advantage of having two cells in the same block
is nullified since there is no guarantee that they will be near each other in
the final placement. Thus, these misshapen blocks place a greater burden
on the place-and-route software to meet routing specifications and timing
specifications on critical paths.
In response to these flaws, we have developed two practical floorplanners. The Hybrid-BF algorithm uses a combination of HCST and the
Breadth First (BF) traversal technique. In addition to guaranteeing minimum-area, Hybrid-BF results in better-shaped flexible blocks and places
them so that they respect their seed locations. The Hybrid-MBF algorithm
uses a Modified Breadth First (MBF) traversal in conjunction with HCST.
It results in better-shaped blocks than Hybrid-BF. Both algorithms require
O ~ g 2 ! time in the worst case, where g is the number of available grid
squares. Optimized implementations of these two algorithms required less
than two seconds on a SUN SPARCstation 10 for each of our industrial
examples.
In Section 2, we formulate the problem and describe the data and input
representation. Section 3 provides the theoretical framework for the HCST
traversal technique and shows that it results in a minimum-area floorplan.
Section 4 describes the Hybrid-BF and Hybrid-MBF algorithms. Section 5
evaluates the quality of the floorplans generated by Hybrid-BF and HybridMBF algorithms and compares it to some of our previous work. This section
also describes six implementations of each algorithm and presents the run
times for each implementation. Section 6 concludes the paper.
2. THE MBC FLOORPLANNING PROBLEM
2.1 Data Representation
We use a data representation in which a uniform grid is imposed on the
chip plane. Each block (fixed or flexible) is denoted by a unique integer. A
grid square is labeled by the integer corresponding to the block that
occupies that square. This ensures that two blocks will never overlap. A
block i is the geometric union of all grid squares labeled i .
2.2 Problem Formulation
Given the grid representation of a floorplan boundary of arbitrary rectilinear shape within which fixed blocks and seeds have been assigned locations
with one seed per flexible block, obtain a floorplan which partitions the
available area (i.e., the grid squares not occupied by fixed blocks) among
the flexible blocks. Each flexible block is allowed to take any rectilinear
shape. The solution should attempt to satisfy the following objectives:
(1) [C]onnectivity: There must be a path between any two grid squares
belonging to the same flexible block, such that neighboring grid squares
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on the path have a side in common and that all grid squares on the path
belong to the flexible block.
(2) [A]rea: We state this objective as follows:
if (available area $ total area required by the flexible blocks)
then a floorplan solution exists.
Clearly, if less area is provided than is required by the flexible blocks,
the problem cannot be solved. Henceforth, we will assume that the area
provided is exactly equal to the sum of the areas of the flexible blocks,
in which case the objective reduces to a minimum-area objective.
(3) [P]roximity: Minimize the mean of the radii of all flexible blocks,
where the radius of a flexible block is defined as the maximum
Manhattan distance of the shortest path between a seed and any grid
square of the flexible block, such that all grid squares on the shortest
path belong to that flexible block. This objective attempts to force the
seed to be at the “center” of the flexible block.
(4) [S]ides: Minimize the number of sides in each flexible block to obtain
better-shaped blocks.
An Example. Figure 1 shows a floorplan grid, four preplaced fixed
blocks, and the seeds corresponding to the four flexible blocks. The seed
locations are denoted by the integers 1– 4 in the figure. The areas of flexible
blocks 1, 2, 3, and 4 are assumed to be 150, 120, 110, and 90, respectively
(all areas are in terms of number of grid squares). Notice that the available
area is 470 5 150 1 120 1 110 1 90.1
3. THE PURE HCST ALGORITHM
It is trivial to satisfy criteria A alone: assuming that the number of
available grid squares exceeds or equals the number of grid squares
required by the flexible blocks, arbitrarily assign the requisite number of
grid squares to each flexible block. If the number of available grid squares
is g , then an algorithm to perform this task requires O ~ g ! time.
Next, we will show that satisfying objectives A and C simultaneously is
nontrivial.
LEMMA 1. There exist problem instances for which it is impossible to
construct a floorplan that satisfies objectives A and C simultaneously.
PROOF. The example of Figure 2 shows such a problem instance, proving
the lemma: the area available for flexible blocks is 23 units. Let there be
two flexible blocks f 1 and f 2 of areas 11 and 12 units respectively, which are
to be placed in the available area. No allocation of grid squares to flexible
blocks is possible such that f 1 and f 2 are both connected. e
1
The boundary of the floorplan was made L-shaped so that the available area is equal to the
required area.
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1
2
4
3
Fig. 1. An example of an input to the MBC floorplanning problem. The areas of flexible
blocks 1, 2, 3, and 4 are assumed to be 150, 120, 110, and 90, respectively.
In view of this, we add some restrictions to the problem instance that
would guarantee that criteria A and C will be satisfied. The first restriction
is to require the available grid squares to be connected. The second (called
the even restriction) is as follows: let any corner of the layout boundary
denote the origin (0,0). Then, both coordinates of every corner of the layout
and of every fixed block must be even.
An example of a layout that satisfies this property is shown in Figure
3(a). The fixed blocks are shaded. If an input grid does not satisfy the even
criterion, we increase the resolution of the grid by a factor of 2 in each
dimension. Any corner ~ i, j ! of the floorplan boundary or fixed block in the
original grid becomes ~ 2i, 2j ! in the new grid and the even restriction is
immediately satisfied.
THEOREM 1. Any problem instance which satisfies the even criterion and
all of whose available grid squares are connected satisfies A and C .
PROOF. Since the coordinates of each corner of the layout and every
fixed block are both even, it is possible to obtain a new grid in which 2 3 2
blocks are replaced by 1 3 1 squares. Figure 3(b) is the modified grid
corresponding to the grid in Figure 3(a). Think of this modified grid as the
connected graph G whose vertices correspond to available grid squares and
whose edges join neighboring squares. Consider any spanning tree of G .
The solid lines of Figure 3(c) delineate a spanning tree for the grid of
Figure 3(b). Traverse the spanning tree starting at the (arbitrarily chosen)
ACM Transactions on Design Automation of Electronic Systems, Vol. 5, No. 1, January 2000.
Flexible, Rectilinear Blocks in Mixed Block and Cell Designs
FIXED BLOCK
Fig. 2.
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FIXED BLOCK
Illustration of Lemma 1.
solid circle, as shown by the dashed line in Figure 3(c). This “twice around
the tree” tour corresponds to a Hamiltonian cycle in the original grid as
shown by the dashed line in Figure 3(a). More precisely, the four quarters
of each vertex in the grid graph correspond to the 4 grid squares in each
2 3 2 block of the original grid.
Finally, we note that a Hamiltonian cycle of length L is sufficient to place
any set of flexible blocks with arbitrary areas whose sum is less than or
equal to L , such that each flexible block is connected. This is done as
follows: choose any flexible block; let its area be a i . Then, traverse the grid
squares in the Hamiltonian cycle until a i squares have been visited. Repeat
with the remaining flexible blocks by continuing to traverse the rest of the
Hamiltonian cycle from the point at which we stopped. e
Creating the spanning tree and traversing the Hamiltonian cycle require
O ~ g ! time, which is optimal. Note that pure HCST makes no attempt to
ensure that the squares occupied by a flexible block are near (or even
include) its seed. Rather, it results in a floorplan in which some flexible
blocks (e.g., 1, 4, and 6) are long snakelike structures (Figure 3(d)).
4. THE HYBRID-BF ALGORITHM
If proximity was the only criterion being considered, we could use a
Breadth-First (BF) traversal technique on the grid to place a flexible block.
Such a traversal starts at the seed of a flexible block and occupies grid
squares (traverses grid graph vertices) until the requisite number of grid
squares (equal to the area of the flexible block) are occupied.
Example: Consider the grid and seed square of Figure 4(a). Assume that
the flexible block corresponding to this seed must occupy 32 squares.
Figure 4(b) shows the placement of the flexible block assuming that the BF
algorithm traverses the neighbors of a square in the order N(orth), E(ast),
S(outh), W(est). Numbers indicate the order in which squares are traversed
(the seed square is assigned the number 0).
LEMMA 2. Given a connected grid, a seed square, and the number of
squares to be occupied, the BF algorithm computes a connected flexible block
with minimum radius.
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(c)
(d)
Fig. 3. (a) Layout satisfying the conditions of Theorem 1. (b) Modified grid. (c) Spanning tree
for modified grid. (d) Floorplan obtained by using HCST assuming that flexible blocks 1,
. . . , 6 require 58, 32, 37, 53, 40, and 68 squares, respectively.
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Fig. 4.
(b)
(c)
Placement of a flexible block using the (b) BF. (c) MBF algorithm.
PROOF. Connectivity follows from the definition of BF—a square is
traversed only if it is adjacent to a square that was traversed earlier. The
minimum radius property follows from the observation that BF occupies all
squares that are a distance i from the seed square before occupying any
square at distance i 1 1 or greater. e
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This suggests an algorithm for floorplanning flexible blocks: place flexible blocks one by one using the BF algorithm outlined above. Note that this
does not guarantee that the mean radius over all flexible blocks is minimized. Furthermore, the grid does not necessarily remain connected. This
could make it impossible to place the remaining flexible blocks. For
example, Figure 4(b) shows that the grid has been disconnected into two
components, one of which contains 5 squares. If there is no set of flexible
blocks in the input whose areas add up to 5, then it is impossible to achieve
connectivity. Finally, the number of sides of each flexible block is likely to
be very high because of the staircase boundaries that are formed when BF
is used.
We propose to reduce the number of sides per flexible block by employing
a modified-BF (MBF) traversal: the MBF traversal attempts to traverse the
N, E, S, and W neighbors of a square followed by its NE, SE, SW, and NW
neighbors. An attempt to traverse a NE neighbor is made only if either its
N or E neighbor was successfully traversed. This is done to ensure that the
original square and its NE neighbor are connected in the final floorplan.
Similar precautions are taken before traversing SE, SW, and NW neighbors. The placement of the flexible block of Figure 4(a) using the MBF
technique results in Figure 4(c). The radius of the block is 5 and the
number of sides is 10, relative to the BF technique where these quantities
are 4 and 24, respectively.
In the remainder of this section, we propose a hybrid algorithm which
combines the BF technique with the HCST technique. Unlike pure HCST,
we do not initially require the grid to satisfy the even criterion. If
necessary, we will double the resolution of portions of the grid. The
algorithm places flexible blocks one at a time. As before, a flexible block is
placed in BFS order by occupying squares starting with the seed. The
difference is that the algorithm verifies that the occupation of a grid square
will not disconnect the grid. The square is occupied if and only if it passes
this test. The BF algorithm guarantees that a flexible block is not placed if
and only if there is no available square adjacent to the flexible block that
does not disconnect the grid. In the event that a flexible block cannot be
placed by BF, the algorithm uses the HCST technique to place the rest of
the flexible block and to start with the next flexible block. The algorithm is
outlined below:
Algorithm Hybrid-BF
1.
for each flexible block F {
2.
s 5 seed square of F ;
3.
while F is not placed {
4.
if (s does not disconnect the grid G of available squares) {
5.
Assign s to F ;
6.
repeat
7.
let s 5 next available square in BF order;
8.
until (s does not disconnect G or no squares available)
9.
}
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A snapshot during the execution of Hybrid.
else { // s disconnects the available squares
Let C be the union of s and all connected components (except the
largest) of the disconnected grid resulting from the occupation of s ;
if (#squares in C # # remaining squares of F )
assign all squares in C to F
else {
Double the grid resolution of C
Use HCST to fill C with all the squares in F followed by squares from
other flexible blocks
Let F 5 the flexible block that is currently being processed.
} // end if
Let s 5 an available neighbor of s in the original grid.
} // end else
} // end of while
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
2
}
We now illustrate the algorithm by running it on the grid of Figure 5.
Flexible blocks 1– 4 have already been placed, and flexible block 5 is being
placed. The only available square s is in position (1, 6) of the grid. This
square, which is obtained in line 7, would disconnect the available grid
squares into two parts, one containing 5 squares, the other containing 50
squares. This causes the algorithm to enter the else clause of line 10 in the
next iteration of the while loop. C (line 11) consists of s and the 5 squares
above it. In general, the grid may be broken into 2, 3, or 4 components. In
the latter two cases, C is the union of the 2 (or 3) smallest components and
s . Assume that 4 squares of block 5 remain to be assigned. This causes the
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else clause of line 14 to be executed. Each square of C is split into four
squares. Since 4 squares of flexible block 5 remain to be assigned in the
original grid, 16 squares of flexible block 5 remain to be assigned in the
new grid. After HCST runs out of 5s (line 16), it begins placing squares
from some other flexible block(s) as shown in the magnified diagram on the
left. After C is filled, we revert to the original grid and continue to place
squares from the current block (number 6) starting from the available
square adjacent to s (1, 7) using the BF algorithm.
THEOREM 2. Hybrid-BF satisfies criteria A and C in a floorplan whose
available grid squares are initially connected.
PROOF. We show that at any time in the algorithm, (1) the flexible
blocks (or their portions) that have been placed are connected and (2) the
available squares (if any) are connected. Clearly, this is true at the
beginning. At the end of the algorithm, when all flexible blocks are placed,
this statement implies that all flexible blocks are connected.
The BF traversal is used to fill squares of a flexible block f 1 up to the
point that a square s is reached (line 10), whose occupation would disconnect the available grid into 2– 4 connected components. From Lemma 2, the
squares that have so far been assigned to f 1 are connected. Also, at this
point the available squares are connected. Lines 11–20 fill squares in C :
the union of $ s % and all but one of the 2– 4 connected components. Note that
C is also connected. C ’s resolution is doubled to satisfy the even criterion.
Next, Pure HCST fills squares in C starting from a sub-square of s that is
adjacent to the f 1 -occupied square from which the BF-traversal attempted
to fill s and ending at a sub-square of s that is adjacent to remaining
connected component that was not included in C . Note that the start and
end sub-squares should be adjacent to each other to facilitate HCST.
Theorem 1 ensures that the flexible blocks that were completely placed in
C are connected. The choice of the start square ensures that f 1 is connected.
The choice of the end square ensures that the flexible block that was being
placed when HCST ran out of squares in C will be connected when the BF
traversal resumes in the next iteration of the while loop. The available
grid is the connected component that was not included in C , and is
connected, by definition. e
O ~ g ! time is required to perform the connectivity check of lines 4 and 8.
A grid square is checked for connectivity at most 4 times (once for each
neighbor). Since there are g grid squares, the worst case complexity of the
BF component of our algorithm is O ~ g 2 ! . The HCST component of the
algorithm is linear in the number of grid squares and does not increase the
time complexity. The Hybrid-MBF algorithm is obtained by replacing the
BF traversal implied in Line 7 with the MBF traversal. The proof of
correctness with respect to Objectives A and C , and the time complexity are
identical to that for Hybrid-BF.
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Table I.
Floorplan
Input Data
No of Fixed Blocks No of Flexible Blocks
FP1
FP2
FP3
FP4
4
4
6
8
4
5
6
8
Area(Fixed)
Area(Flexible)
77
104
137
445
470
448
1605
1890
5. EXPERIMENTAL RESULTS
Six variations of both hybrid algorithms were implemented in C11 and
run on a SPARCstation10. The six variations only differ from each other in
their implementations and therefore their run times, but all compute the
same floorplan. For data, we used 4 industrial floorplans (Table I) [Mehta
et al. 1995].
Quality of Floorplans
The quality metrics for each of the four floorplans are shown in Table II.
A opt is the optimal area obtained by both algorithms. A COLONIZE is the area
we obtained by using an iterative algorithm with an unbounded time
complexity [Mehta et al. 1995]. P BF and P MBF are the proximities obtained
by the BF- and MBF-based algorithms, respectively. P LB is a lower bound
on the proximity for a given set of flexible blocks. It is computed by
assuming that each flexible block has minimum radius. The minimum
radius of a flexible block is the radius obtained when a BF algorithm is
used to place the block in an infinitely large grid all of whose squares are
available. S BF and S MBF are the average number of sides of all flexible
blocks obtained by the BF- and MBF-based algorithms, respectively. The
lower bound is 4, when all flexible blocks are rectangles. As expected, there
is a significant drop in the number of sides when MBF is used instead of
BF. This is accompanied by a slight increase in proximity. In one case, the
proximity improved when MBF was used.
Figures 6 and 7 show some of the floorplans obtained by our algorithms.
Figures 6(a) and 7(a) correspond to the input of Figure 1. The integers
associated with the flexible blocks indicate the order in which they were
processed, while their position within each flexible block denotes the
location of the seed for that flexible block. In all the cases tried, the
algorithm never had to execute the else clause of line 14 of the hybrid
algorithm. This can be attributed to the large ratio of standard cell area to
fixed block area in the floorplans used in our experiments. Since pure
HCST ignores proximity and side minimization, Hybrid-BF and HybridMBF can be expected to result in poor proximity and side minimization for
circuits with low standard cell area/fixed block area ratios.
An important observation resulting from our experiments is that the
relative positions of fixed blocks and seeds in the problem instance are
crucial to succeeding with respect to the proximity objective. A poor initial
placement (e.g., Figure 6(a)) in which seeds and fixed blocks are clustered
ACM Transactions on Design Automation of Electronic Systems, Vol. 5, No. 1, January 2000.
Flexible, Rectilinear Blocks in Mixed Block and Cell Designs
Table II.
93
•
Quality of MBC Floorplanning Algorithms
Floorplan
A opt
A COLONIZE
P BF
P MBF
P LB
S BF
S MBF
FP1
FP2
FP3
FP4
547
552
1742
2335
576
600
1764
2500
23.0
15.6
20.9
29.0
23.75
15.40
23.40
30.90
7.8
6.6
9.8
11.0
38.0
36.8
61.5
72.0
19.5
15.2
18.75
20.0
1
1
2
2
4
3
3
4
(a)
Fig. 6.
(b)
Floorplans generated by the hybrid BF-based algorithm.
1
1
2
2
4
3
3
4
(a)
Fig. 7.
(b)
Floorplans generated by the hybrid MBF-based algorithm.
together will result in poor proximity, whereas a better initial placement
(Figure 6(b)) gives better proximity results. Average radii for flexible blocks
in Figure 6(a) and Figure 6(b) were 23 and 14.25, respectively. Figure 7
shows the floorplans obtained by using MBF instead of BF on the same
inputs as Figure 6. Average radii for Figures 7(a) and 7(b) were 23.75 and
15, respectively.
ACM Transactions on Design Automation of Electronic Systems, Vol. 5, No. 1, January 2000.
94
•
D. P. Mehta and N. Sherwani
Implementation Issues
A check was added which ensures that the seed square of a flexible block is
never filled by another flexible block. This results in the narrow paths often
used to connect the seed to the rest of the block in Figure 6(a) (for blocks 2,
3, and 4).
We experimented with three variations of the BF and MBF algorithms.
The three variations use different algorithms to check that the occupation
of a grid square does not disconnect the available grid. Since this check is
performed each time a grid square is being allotted to a flexible block, its
impact on run times is quite significant. In the first variation (V1), a
depth-first algorithm which starts at any available square, is used to
traverse the connected component containing that square. The number of
squares in that connected component equals the number of squares available in the entire floorplan if and only if the available grid squares are
connected. The second and third variations utilize the following observation.
LEMMA 3. Occupying an available square, c disconnects an originally
connected grid if and only if there is no path between at least one pair of c ’s
(at most 4) available neighbors.
PROOF. if: Clearly, if there is no path between a pair of squares, the grid
is disconnected.
only if: Suppose that the occupation of grid square c disconnects the grid,
but there is a path between each pair of c ’s available neighbors. Consider
the grid prior to c ’s occupation. Since the grid was originally connected,
there is a path between each square and c . Consider the shortest such path.
This path must go through one of c ’s available neighbors and it cannot go
through two of c ’s neighbors (otherwise it would not be a shortest path).
After c ’s occupation, there will still be a path between each square and one
of c ’s neighbors (i.e., the shortest path in the original grid without the last
edge). But, since each pair of c ’s neighbors are connected, all grid squares
are connected to each other—so the grid is not disconnected, which is a
contradiction. e
Figure 8 illustrates this observation. Square c 2 disconnects the available
grid while squares c 1 and c 3 do not. Notice that there is no path between
the available neighbors w 2 and e 2 of c 2 , once c 2 is occupied. On the other
hand, there continue to be paths between any pair of (n 3 , w 3 , s 3 ) after c 3 is
occupied. c 1 has only one available neighbor, hence its occupation does not
disconnect the grid.
The second variation ~ V2 ! takes advantage of this by counting c ’s
available neighbors. If this quantity is 1 (e.g., for c 1 in Figure 8), then c
does not disconnect the grid, and it is not necessary to run the DF
algorithm to check for connectivity. The third variation ( V3 ) differs from
the second when c has two or more available neighbors. In this case, a BF
ACM Transactions on Design Automation of Electronic Systems, Vol. 5, No. 1, January 2000.
Flexible, Rectilinear Blocks in Mixed Block and Cell Designs
Fig. 8.
•
95
Illustration of Lemma 3.
traversal starting at one of c ’s available neighbors is used. If this traversal
visits all of the remaining available neighbors of c , then c does not
disconnect the grid and the BF traversal is immediately halted. In practice,
the neighbors of c are often connected by short paths and the BFS traversal
halts after visiting very few squares. In contrast, the DF algorithm used in
the first two variations always traverses all available squares.
The BF and DF algorithms mentioned above traverse the grid by marking all the squares visited. After the traversal is concluded, it is necessary
to reset all the marked values in the grid.2 For each of the three variations
we experimented with two methods for resetting the grid. In the first ~ M1 ! ,
a DF traversal which only traverses marked squares is used. In the second
~ M2 ! , all squares in the two-dimensional array used to store the grid are
traversed. Table III shows the run times for each of the six variations of the
hybrid BF and MBF algorithms for the four floorplans described in Table I.
We summarize the results below: (1) In general, the hybrid BF algorithm is
faster than the hybrid MBF algorithm. (2) As expected, variation V 3 is
faster than V 2 which is faster than V 1 . V 1 is slower than V 2 by a factor of
1.14 –1.32, and is slower than V 3 by a factor of 3.27–27.39. (3) M 1 takes
more time than M 2 for variations V 1 and V 2 , while M 2 takes more time for
variation V 3 .
6. CONCLUSIONS
In the preceding sections, we have outlined two practical hybrid algorithms
for floorplanning flexible blocks in MBC designs. These algorithms employ
2
S. Hauck proposed an elegant technique that makes it possible to avoid this step: Suppose
that an unsigned integer is used to represent the contents of the grid squares. Then, we
reserve the highest integers in its range to denote fixed and flexible blocks. Let the remaining
integers be [0, maxint]. Initially, all available squares contain 0. When grid connectivity is
checked for the i th time ~ i $ 1 ! , integer i is used to mark squares; squares containing values
less than i are considered to be available in this iteration. It is not necessary to reset squares
marked i , since they will be treated as available in subsequent connectivity checks. After all
maxint integers are used, it will, however, be necessary to reset all squares with integers in
[1, maxint]. This practically removes the overhead of unmarking squares at the end of each
connectivity check.
ACM Transactions on Design Automation of Electronic Systems, Vol. 5, No. 1, January 2000.
96
D. P. Mehta and N. Sherwani
•
Table III.
~ V, M !
~ V 1,
~ V 1,
~ V 2,
~ V 2,
~ V 3,
~ V 3,
M 1!
M 2!
M 1!
M 2!
M 1!
M 2!
MBC Floorplanning Algorithm Run Times (in seconds)
FP1
FP2
FP3
FP4
MBF
BF
MBF
BF
MBF
BF
MBF
BF
1.31
0.73
0.98
0.56
0.21
0.20
1.17
0.66
0.90
0.50
0.12
0.14
0.87
0.49
0.66
0.37
0.10
0.11
0.89
0.49
0.74
0.41
0.15
0.15
15.99
8.72
13.72
7.49
0.97
1.19
14.44
7.90
12.69
6.92
0.89
1.09
22.02
12.11
18.57
10.21
1.58
1.81
19.45
10.69
16.68
9.16
0.71
1.20
the new HCST traversal technique introduced in this paper in combination
with traditional graph traversal techniques to achieve minimum-area floorplans. Both algorithms attempt to locate flexible blocks at the specified
seed positions and both require O ~ g 2 ! time, where g is the number of grid
squares. The MBF-based algorithm computes floorplans with significantly
better shapes than the BF-based algorithms.
The quality of a floorplan is closely tied to the quality of the input which
consists of seed and fixed block positions and area estimates for flexible
blocks. The area estimates for flexible blocks must take routing area into
consideration. If the floorplan quality is unsatisfactory (e.g., timing/routing
specifications cannot be met in subsequent steps), it may be necessary to
rerun the floorplanner with better input data. The fast speed of our
algorithms makes it possible to rerun the floorplanner on improved input
specifications with minimal impact to the design cycle time. Studies on the
impact of input specifications on the floorplanner would be useful. In this
regard, it will also be useful to study more formally the impact of various
shapes on routability and timing.
ACKNOWLEDGMENTS
The authors thank one of the referees for pointing out a flaw in one of our
results, and Prof Scott Hauck for a suggestion on improving the implementation.
REFERENCES
APTE, J. AND KEDEM, G. 1990. Heuristic algorithms for combined standard cell and macro
block layouts. In Proceedings of the Sixth MIT Conference on Advanced Research in VLSI
(AUSCRYPT ’90, MIT, Boston, MA, 1990), W. J. Dally, Ed. MIT Press, Cambridge, MA,
367–385.
CHONG, K. AND SAHNI, S. 1993. Optimal realizations of floorplans. IEEE Trans. Comput.Aided Des. 12 (June 1993), 793– 801.
DAI, W., ESCHERMANN, B., KUH, E., AND PEDRAM, M. 1989. Hierarchical placement and
floorplanning in BEAR. IEEE Trans. Comput.-Aided Des. 8, 1335–1349.
DASGUPTA, P. S., SUR-KOLAY, S., AND BHATTACHARYA, B. B. 1995. A unified approach to
topology generation and area optimization of general floorplans. In Proceedings of the 1995
IEEE/ACM International Conference on Computer-Aided Design (ICCAD-95, San Jose, CA,
Nov. 5–9), R. Rudell, Ed. IEEE Computer Society Press, Los Alamitos, CA, 712–715.
ACM Transactions on Design Automation of Electronic Systems, Vol. 5, No. 1, January 2000.
Flexible, Rectilinear Blocks in Mixed Block and Cell Designs
•
97
KOZMINSKI, K. A. AND KINNEN, E.
1988.
Rectangular dualization and rectangular
dissections. IEEE Trans. Circ. Syst. 35 (Nov. 1988), 1401–1416.
LEE, T.-C. 1993. A bounded 2D contour searching algorithm for floorplan design with
arbitrarily shaped rectilinear and soft modules. In Proceedings of the 30th International
Conference on Design Automation (DAC ’93, Dallas, TX, June 14 –18), A. E. Dunlop, Ed.
ACM Press, New York, NY, 525–530.
MEHTA, D. P., SHANBHAG, A., AND SHERWANI, N. A. 1995. A new approach for floorplanning in
MBC designs. Technical Report No. 95-02. Space Institute, Univ. Tennessee, Knoxville,
TN.
NAKATAKE, S., FUJIYOSHI, K., MURATA, H., AND KAJITANI, Y. 1996. Module placement on
BSG-structure and IC layout applications. In Proceedings of the 1996 IEEE/ACM International Conference on Computer-Aided Design (ICCAD ’96, San Jose, CA, Nov. 10 –14, 1996),
R. A. Rutenbar and R. H. J. M. Otten, Eds. IEEE Computer Society Press, Los Alamitos,
CA, 484 – 491.
OTTEN, R. 1982. Automatic floorplan design. In Proceedings of the 19th ACM/IEEE
Conference on Design Automation ACM Press, New York, NY, 261–267.
OTTEN, R. 1983. Efficient floorplan optimization. In Proceedings of the International
Conference on Computer Aided Design 499 –503.
PAN, P. AND LIU, C. L. 1995. Area minimization for floorplans. IEEE Trans. Comput.-Aided
Des. 14 (Jan.), 123–132.
PAN, P., SHI, W., AND LIU, C. L. 1994. Area minimization for hierarchical floorplans. In
Proceedings of the 1994 IEEE/ACM International Conference on Computer-Aided Design
(ICCAD ’94, San Jose, CA, Nov. 6 –10, 1994), J. A. G. Jess and R. Rudell, Eds. IEEE
Computer Society Press, Los Alamitos, CA, 436 – 440.
PUTATUNDA, R., SMITH, D., STEBNISKY, M., PUSCHAK, C., AND PATENT, P. 1988. VITAL: Fully
automatic placement strategies for very large semicustom designs. In Proceedings of the
International Conference on Computer Design (ICCD ’88) 434 – 439.
SARRAFZADEH, M. AND WONG, C. K.
1996.
An Introduction to VLSI Physical
Design. McGraw-Hill, Inc., New York, NY.
SHERWANI, N. 1995. Algorithms for VLSI Physical Design Automation. Kluwer Academic
Publishers, Hingham, MA.
STOCKEMEYER, L. 1983. Optimal orientation of cells in slicing floorplan designs. Inf. Control
57, 2, 91–101.
SUTHANTAVIBUL, S., SHRAGOWITZ, E., AND ROSEN, J. B. 1991. An analytic approach to floorplan
design and optimization. IEEE Trans. Comput.-Aided Des. 10 (June 1991), 761–769.
UPTON, M., SAMII, K., AND SUGIYAMA, S. 1990. Integrated placement for mixed macro cell and
standard cell designs. In Proceedings of the ACM/IEEE Conference on Design Automation
(DAC ’90, Orlando, FL, June 24-28), R. C. Smith, Ed. ACM Press, New York, NY, 32–35.
VIJAYAN, G. AND TSAY, R. 1991. A new method for floorplanning using topological constraint
reduction. IEEE Trans. Comput.-Aided Des. 10 (Dec. 1991), 1494 –1501.
WANG, T. AND WONG, D. 1988. Optimal floorplanning area minimization. IEEE Trans.
Comput.-Aided Des. CAD-7, 11 (Nov.), 992–1002.
WONG, D. F. AND LIU, C. L. 1989. Floorplan design of VLSI circuits. Algorithmica 4, 263–291.
YEAP, K. AND SARRAFZADEH, M. 1993. Floorplanning by graph dualization: 2-concave
rectilinear modules. SIAM J. Comput. 22 (June 1993), 500 –526.
Received: April 1996;
revised: November 1997;
accepted: July 1998
ACM Transactions on Design Automation of Electronic Systems, Vol. 5, No. 1, January 2000.